[Q] Email not persistent on my Note 3

There was only one thread near what I am experiencing with no solution so apologies now if I am redundant to another thread I don't see!

Background:
Device is Note 3 purchase within this week. Using stock email client at the moment. Configured a new account on my Note 3 to sync to my company email through a third party pop mail server (@mail on pair.com) used for remote access to email.

My Issue:
email I receive on my Note 3 while remote does not persist on the device over time. I presume when I access those emails while at my office computer, the third party server at pair.com deletes them for remote access. Therefor, those emails are not really being downloaded to my device, rather, just displaying what is on the third party server at each sync point. I have a business need to keep remotely accessed email on my device until I choose to remove them. I have gone through every possible setting on my device to no avail. I am not a company admin on the pair.com site so I am not sure what settings are there that drive behavior with respect to my device client. If need be, I do know the site admin.

My question:
Is there something I'm missing here? Has anyone had this requirement for their device and knows how to configure settings to force true email downloads to the device, making them autonomous to the device for managing locally (i.e. Keep / Delete)?

BTW, I will be abandoning the stock client in favor of K-9 for other reasons - will I run into the same problem with K-9 settings/behavior?

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